Simplify and expand the following expression: $ \dfrac{n + 7}{n - 10}+\dfrac{n}{3n + 8} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(n - 10)(3n + 8)$ Multiply the first term by $\dfrac{3n + 8}{3n + 8}$ $ \begin{align*} \dfrac{n + 7}{n - 10} \times \dfrac{3n + 8}{3n + 8} & = \dfrac{(n + 7)(3n + 8)}{(n - 10)(3n + 8)} \\ & = \dfrac{3n^2 + 29n + 56}{(n - 10)(3n + 8)}\end{align*} $ Multiply the second term by $\dfrac{n - 10}{n - 10}$ $ \begin{align*} \dfrac{n}{3n + 8} \times \dfrac{n - 10}{n - 10} & = \dfrac{(n)(n - 10)}{(3n + 8)(n - 10)} \\ & = \dfrac{n^2 - 10n}{(3n + 8)(n - 10)}\end{align*} $ Now we have: $ = \dfrac{3n^2 + 29n + 56}{(n - 10)(3n + 8)} + \dfrac{n^2 - 10n}{(3n + 8)(n - 10)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{3n^2 + 29n + 56 + n^2 - 10n}{(n - 10)(3n + 8)} $ $ = \dfrac{4n^2 + 19n + 56}{(n - 10)(3n + 8)}$ Expand the denominator: $ = \dfrac{4n^2 + 19n + 56}{3n^2 - 22n - 80}$